We're asked, what is the
greatest common divisor of 20 and 40? And they just say,
another way to say this is the GCD, or greatest
common divisor, of 20 of 40 is equal to question mark. And greatest common divisor
sounds like a very fancy term, but it's really
just saying, what is the largest number that is
divisible into both 20 and 40? Well, this seems like a pretty
straightforward situation, because 20 is actually
divisible into 40. Or another way to
say it is 40 can be divided by 20
without a remainder. So the largest
number that is a-- I guess you could say-- factor of
both 20 and 40 is actually 20. 20 is 20 times 1,
and 40 is 20 times 2. So in this situation,
we don't even have to break out our paper. We can just write 20. Let's do a couple more of these. So we're asked, what is
the greatest common divisor of 10 and 7? So let's now break out
our paper for this. So our greatest common
divisor of 10 and 7. So let me write that down. So we have 10. We want to think about what
is our GCD of 10 and 7? And there's two ways that
you can approach this. One way, you could literally
list all of the factors-- not prime factors, just
regular factors-- of each of these numbers and figure
out which one is greater or what is the largest
factor of both. So, for example, you could
say, well, I got a 10, and 10 can be expressed
1 times 10 or 2 times 5. 1, 2, 5, and 10. These are all factors of 10. These are all, we could
say, divisors of 10. And sometimes this is called
greatest common factor. Seven-- what are
all of its factors? Well, 7 is prime. It only has two
factors-- 1 and itself. So what is the
greatest common factor? Well, there's only one
common factor here, 1. 1 is the only common factor. So the greatest common
factor of 10 and 7, or the greatest common divisor,
is going to be equal to 1. So let's write that down. 1. Let's do one more. What is the greatest common
divisor of 21 and 30? And this is just another
way of saying that. So 21 and 30 are the two
numbers that we care about. So we want to figure out
the greatest common divisor, and I could have written
greatest common factor, of 21 and 30. So once again, there's
two ways of doing this. And so there's the way I did
the last time where I literally list all the factors. Let me do it that
way really fast. So if I say 21, what
are all the factors? Well, it's 1 and
21, and 3, and 7. I think I've got all of them. And 30 can be written as 1 and
30, 2 and 15, and 3-- actually, I'm going to run out of them. Let me write it this way so
I get a little more space. So 1 and 30. 2 and 15. 3 and 10. And 5 and 6. So here are all of
the factors of 30. And now what are
the common factors? Well, 1 is a common factor. 3 is also a common factor. But what is the
greatest common factor or the greatest common divisor? Well, it is going to be 3. So we could write 3 here. Now, I keep talking
about another technique. Let me show you the
other technique, and that involves the
prime factorization. So if you say the prime
factorization of 21-- well, let's see, it's divisible by 3. It is 3 times 7. And the prime factorization
of 30 is equal to 3 times 10, and 10 is 2 times 5. So what are the
most factors that we can take from both 21 and
30 to make the largest possible numbers? So when you look at the
prime factorization, the only thing that's common
right over here is a 3. And so we would say that
the greatest common factor or the greatest common
divisor of 21 and 30 is 3. If you saw nothing in
common right over here, then you say the greatest
common divisor is one. Let me give you another
interesting example, just so that we can get a
sense of things. So let's say these two
numbers were not 21 and 30, but let's say we care about
the greatest common divisor not of 21, but let's
say of 105 and 30. So if we did the prime
factorization method, it might become a
little clearer now. Actually figuring out, hey,
what are all the factors of 105 might be a little bit
of a pain, but if you do a prime factorization,
you'd say, well, let's see, 105-- it's divisible
by 5, definitely. So it's 5 times 21,
and 21 is 3 times 7. So the prime
factorization of 105 is equal to-- if I write them
in increasing order-- 3 times 5 times 7. The prime factorization of
30, we already figured out is 30 is equal to
2 times 3 times 5. So what's the most
number of factors or prime factors that
they have in common? Well, these two both have a
3, and they both have a 5. So the greatest common factor
or greatest common divisor is going to be a
product of these two. In this situation,
the GCD of 105 and 30 is 3 times 5, is equal to 15. So you could do it either way. You could just list out the
traditional divisors or factors and, say, figure
out which of those is common and is the greatest. Or you can break it down
into its core constituencies, its prime factors,
and then figure out what is the largest set
of common prime factors, and the product
of those is going to be your greatest
common factor. It's the largest number that
is divisible into both numbers.